Facts used

Proof

Given: A group of prime power order, expressed as an internal direct product of nontrivial subgroups and .

To prove: is not characteristic in .

Proof: Let be a maximal subgroup of . Then, is normal and is cyclic of prime order. Let be a subgroup of order contained in the center of . Thus, we can construct a surjective homomorphism with kernel .

Now, consider the map:

.

This map is clearly a bijection from to . It is a homomorphism because . Thus, is an automorphism of . Moreover, does not leave invariant, showing that is not characteristic.